Size-biased permutation of a finite sequence with independent and identically distributed terms

Iterated Decomposition of Biased Permutations Via New Bounds on the Spectral Gap of Markov Chains

The spectral gap of a Markov chain can be bounded by the spectral gaps of constituent "restriction" chains and a "projection" chain, and the strength of such a bound is the content of various decomposition theorems. In this paper, we introduce a new parameter that allows us to improve upon these bounds. We further define a notion of orthogonality between the restriction chains and "complementary" restriction chains. This leads to a new Complementary Decomposition theorem, which does not require analyzing the projection chain. For $ε$-orthogonal chains, this theorem may be iterated $O(1/ε)$ tim…

Ordered and size-biased frequencies in GEM and Gibbs models for species sampling

We describe the distribution of frequencies ordered by sample values in a random sample of size $n$ from the two parameter GEM$(α,θ)$ random discrete distribution on the positive integers. These frequencies are a $($size$-α)$-biased random permutation of the sample frequencies in either ranked order, or in the order of appearance of values in the sampling process. This generalizes a well known identity in distribution due to Donnelly and Tavaré (1986) for $α= 0$ to the case $0 \le α< 1$. This description extends to sampling from Gibbs$(α)$ frequencies obtained by suitable conditioning of the G…

Some remarks on biased recursive trees

The purpose of this paper is to analyze certain statistics of a recently introduced non-uniform random tree model, biased recursive trees. This model is based on constructing a random tree by establishing a correspondence with non-uniform permutations, biased riffle shuffles. The statistics that are treated include the number of nodes with a given number of descendants, the depth of the tree, and the number of branches. The model yields the uniform recursive trees as a certain limit, some new results for the uniform case are obtained as well.

Limit theorems for a class of identically distributed random variables

A new type of stochastic dependence for a sequence of random variables is introduced and studied. Precisely, (X_n)_{n\geq 1} is said to be conditionally identically distributed (c.i.d.), with respect to a filtration (G_n)_{n\geq 0}, if it is adapted to (G_n)_{n\geq 0} and, for each n\geq 0, (X_k)_{k>n} is identically distributed given the past G_n. In case G_0={\varnothing,Ω} and G_n=σ(X_1,...,X_n), a result of Kallenberg implies that (X_n)_{n\geq 1} is exchangeable if and only if it is stationary and c.i.d. After giving some natural examples of nonexchangeable c.i.d. sequences, it is shown th…

On permutations of lacunary series

It is a well known fact that for periodic measurable $f$ and rapidly increasing $(n_k)_{k \geq 1}$ the sequence $(f(n_kx))_{k\ge 1}$ behaves like a sequence of independent, identically distributed random variables. For example, if $f$ is a periodic Lipschitz function, then $(f(2^kx))_{k\ge 1}$ satisfies the central limit theorem, the law of the iterated logarithm and several further limit theorems for i.i.d.\ random variables. Since an i.i.d.\ sequence remains i.i.d.\ after any permutation of its terms, it is natural to expect that the asymptotic properties of lacunary series are also permutat…

Concentration of measures via size biased couplings

Let $Y$ be a nonnegative random variable with mean $μ$ and finite positive variance $σ^2$, and let $Y^s$, defined on the same space as $Y$, have the $Y$ size biased distribution, that is, the distribution characterized by E[Yf(Y)]=μE f(Y^s) for all functions $f$ for which these expectations exist. Under a variety of conditions on the coupling of Y and $Y^s$, including combinations of boundedness and monotonicity, concentration of measure inequalities hold. Examples include the number of relatively ordered subsequences of a random permutation, sliding window statistics including the number of m…

On permutations of Hardy-Littlewood-Pólya sequences

Let ${\cal H}=(q_1, \ldots q_r)$ be a finite set of coprime integers and let $n_1, n_2, \ldots$ denote the multiplicative semigroup generated by $\cal H$ and arranged in increasing order. The distribution of such sequences has been studied intensively in number theory and they have remarkable probabilistic and ergodic properties. For example, the asymptotic properties of the sequence $\{n_kx\}$ are very similar to those of independent, identically distributed random variables; here $\{\cdot \}$ denotes fractional part. However, the behavior of this sequence depends sensitively on the generatin…

de Finetti–Hewitt–Savage定理的推广

Generalizing the de Finetti--Hewitt--Savage theorem

本研究利用非标准分析工具，将de Finetti–Hewitt–Savage定理推广至任意豪斯多夫空间上的Radon分布可交换序列，还给出非标准分析入门介绍及Prokhorov定理的新推广。